Lesson 10-1

Graphing Quadratic Functions

Objectives

• Graph quadratic functions

• Find the equation of the axis of symmetry and the coordinates of the vertex of a parabola

Vocabulary

• Quadrant – • Parabola – • Minimum – • Maximum – • Vertex – • Symmetry – • Axis of symmetry –

Four Step Problem Solving Plan

• Step 1: Explore the Problem

– Identify what information is given (the facts)

– Identify what you are asked to find (the question)

• Step 2: Plan the Solution

– Find an equation the represents the problem

– Let a variable represent what you are looking for

• Step 3: Solve the Problem

– Plug into your equation and solve for the variable

• Step 4: Examine the Solution

– Does your answer make sense?

– Does it fit the facts in the problem?

Example 1

Use a table of values to graph

Graph these ordered pairs and connect them with a smooth curve.

Answer:

x y

–2 –4

–1 0

0 –2

1 –2

2 0

3 4

Example 2

Graph these ordered pairs and connect them with a smooth curve.

Answer:

Use a table of values to graph

x y

–2 –8

–1 0

0 4

1 4

2 0

3 –8

Example 3a

Consider the graph of Write the equation of the axis of symmetry.

In

Equation for the axis of symmetry of a parabola

and

Answer: The equation of the axis of symmetry is

Example 3b&cB. Consider the graph of Find the coordinates of the vertex.

Since the equation of the axis of symmetry is x = –2 and the vertex lies on the axis, the x-coordinate for the vertex is –2.

Original equation

Simplify.

Add.

Answer: The vertex is at (–2, 6).

C. Identify the vertex as a maximum or minimum.

Answer: Since the coefficient of the x2 term is negative, the parabola opens downward and the vertex is a maximum point.

Example 3dGraph the function.

You can use the symmetry of the parabola to help you draw its graph. On a coordinate plane, graph the vertex and the axis of symmetry.

(–2, 6)

Choose a value for x other than –2. For example, choose –1 and find the y-coordinate that satisfies the equation.

Original equation

Example 3d contGraph the function.

(–2, 6)

Graph (–1, 4).(–1, 4)

Since the graph is symmetrical about its axis of symmetry x = –2, you can find another point on the other side of the axis of symmetry. The point at (–1, 4) is 1 unit to the right of the axis. Go 1 unit to the left of the axis and plot the point (–3, 4).

(–3, 4)

Example 3d cont

Graph the function.(–2, 6)

Repeat this for several other points.

(–1, 4)

Then sketch the parabola.

(–3, 4)

(0, –2)(–4, –2)

Example 4

Multiple-Choice Test ItemWhich is the graph of

A B

C D

Example 4 cont

Solve the Test ItemFind the axis of symmetry of the graph

Equation for the axis of symmetry

and

The axis of symmetry is –1. Look at the graphs. Since only choices C and D have this as their axis of symmetry, you can eliminate choices A and B. Since the coefficient of the x2 term is negative, the graph opens downward. Eliminate choice C.

Answer: D

Summary & Homework

• Summary:– The standard form of a quadratic function is

y = ax2 + bx + c– Complete a table of values to graph a quadratic

function– The equation of the axis of symmetry for the graph

of y = ax2 + bx + c, where a ≠ 0 is x = (-b/2a)– The vertex of a parabola is on the axis of symmetry

• Homework: – pg